This is my understanding. Pls advise if I am correct.
From what I understand, S11 is a measure of the reflection from an antenna. 0dB means that all the power is reflected, hence the matching is not good. -10dB means that 0.1 * incident power is reflected, meaning 0.9*power is accepted by the antenna. Coupled with the fact that most antenna designed are with low loss, the 0.9*power is radiated out. Hence, when we see a dip (V shape) cutting the -10dB level, we normally label the apex as the resonant frequency. However, we can also apply a matching network to the antenna to make sure it is conjugate-matched to 50ohms. If so, we can design antenna with any frequency we want?

You are correct - S11 is a measure of how much power the antenna "accepts". That is, if S11 is -10 dB, then 90% of the energy a receiver sends is not reflected by the antenna - it is either absorbed (as losses) or radiated away.

Antennas are designed to be low loss, however, a good value for S11 (or equivalently, a low value of VSWR) does not necessarily mean the antenna is radiating. Often there are loss mechanisms in place that significantly reduce the efficiency, particularly if the antenna is near any electronics or any other lossy dielectric.

Your last question is interesting: can we use impedance matching to allow the antenna to radiate at whatever frequency we choose? There are some problems with that.

Suppose an antenna such as a dipole is lambda/2 in length at 1 GHz say, and you use impedance matching to tune it to 500 MHz. The issues that arise are:
1) The bandwidth of the antenna at 500MHz will be significantly reduced - that is, if the bandwidth was 10% at 1 GHz, it may only be 1% at 500MHz.
2) The efficiency will be much lower. An impedance matching network will introduce losses (particularly inductors tend to have a lot of loss). And since the antenna is no longer the correct size, the radiation resistance of the antenna will be much lower. Hence, a sizeable fraction of the power will be absorbed as losses and you can expect a significant drop in efficiency.

So in summary, you can't tune an antenna with impedance matching to be arbitrarily low in frequency. That is, you can't take a 1 meter antenna and use impedance matching to reduce the size to 1 centimeter.

Thanks for the compliment. Actually, I have learnt a lot from your informative website which is quite intuitive to understand. Hence, you may find my message similar to your website text.

Q1. Having a good value for S11 (less than -10dB) does not mean that the antenna is radiating? But I thought that is how we find the resonant frequency of an antenna by looking for the V shape dip.

Q2. Is resonant similar to radiating at its best in the antenna context?

Q3. Still quite confused about conjugate-matching and resonance. So, I shall write down what I know. Pls advise if there is anything wrong.
An antenna resonate when its impedance is real, ie no reactance and only real resistance exists. This occurs when the inductive portion cancels with the capacitive portion of the antenna (in the case of PIFA). The reason for an antenna resonating only when its impedance is real is because only then it can be conjugate-matched to 50-ohm transmission line. Conjugate-matching will guarantee the maximum power transfer from the transmission line to the antenna. Maximum power transfer can also be seen as minimum reflection which is indicated by a low S11. So, low S11 implies resonance. To achieve low S11, we can create a matching network between the antenna and the 50 ohm transmission line. So, if the antenna has some reactance, we can use the matching network to cancel the reactance to allow for conjugate-matching, hence low S11, hence resonance can occur.

A1. If we put a 50 Ohm resistor on the end of a coaxial cable, S11 will be -50 dB or so (no reflected power). So we can't conclude the antenna is radiating. The real test is to measure the antenna efficiency (power radiated versus input power) in an anechoic chamber. However, since chamber measurements are sometimes hard to come by and take a while, often in antenna literature you simply see the S11 frequency response. This does give you a good indication of where in frequency the antenna has the potential to radiate; however, a good S11 response does not necessarily mean the antenna is radiating. S11 is still typically used though to show an antenna's response; the underlying assumption is that the losses are not so great.

A2. Resonant is somewhat ambiguous as some antenna engineers use it in varying ways. Resonant can be the frequency of peak efficiency for a given band, or where the impedance is entirely real. The loose definition is "where the antenna is radiating the most power'.